1
00:00:05,270 --> 00:00:10,798
If the discriminant is negative, then the zeros are going to be complex zeros,

2
00:00:10,798 --> 00:00:13,350
they're going to be imaginary zeros.

3
00:00:16,490 --> 00:00:35,467
So speaking of that, let this be any quadratic function. How many zeros are we

4
00:00:35,467 --> 00:00:40,090
guaranteed to have?

5
00:00:41,210 --> 00:00:49,235
Two. What kind? Two complex zeros. They might be real or they might be imaginary,

6
00:00:49,235 --> 00:00:54,090
but we definitely have two complex zeros. Agreed?

7
00:00:54,090 --> 00:01:14,159
What are the two zeros? To find zeros, what do we do? If we plug in x = 0, we get

8
00:01:14,159 --> 00:01:24,070
f(0) = a times 0 squared plus b times 0…

9
00:01:25,930 --> 00:01:46,967
That gives the y-intercept. We want zeros, so plug 0 into f(x), or y. Then how do we

10
00:01:46,967 --> 00:01:48,470
solve?

11
00:01:52,430 --> 00:02:05,927
By factoring, completing the square, or the quadratic formula. This is how we

12
00:02:05,927 --> 00:02:16,970
derived the formula: the two zeros are [−b ± √(b² − 4ac)]/(2a).

13
00:02:18,330 --> 00:02:29,244
Does that make sense? It is abstract because these are variables, but they are all

14
00:02:29,244 --> 00:02:37,230
numbers. These are the two zeros. I could write it this way.

15
00:02:37,860 --> 00:02:43,000
We agree? Okay.

16
00:02:44,560 --> 00:02:51,440
Let me put an absolute value inside the square root to make this correct.

17
00:02:51,960 --> 00:03:07,800
If b² − 4ac is negative, these zeros are imaginary.

18
00:03:09,080 --> 00:03:22,386
We get a square root of a negative number: the square root of its magnitude times

19
00:03:22,386 --> 00:03:26,000
√(−1). And √(−1) is i.

20
00:03:26,000 --> 00:03:38,869
The absolute value makes that negative discriminant positive inside the radical; the

21
00:03:38,869 --> 00:03:41,780
i comes from √(−1).

22
00:03:41,780 --> 00:04:00,348
These are the two zeros: −b/(2a) plus the square-root term. I will keep the absolute

23
00:04:00,348 --> 00:04:03,000
value there.

24
00:04:07,560 --> 00:04:21,380
And −b/(2a) minus the square-root term over 2a.

25
00:04:22,430 --> 00:04:25,110
These are two complex numbers.

26
00:04:32,030 --> 00:04:35,030
What's the only difference between these two?

27
00:04:38,150 --> 00:04:45,730
Plus versus minus. What do we call that? Conjugates.

28
00:04:46,730 --> 00:05:03,090
This is an important fact: if b² − 4ac is less than zero—negative—

29
00:05:03,090 --> 00:05:27,230
the two zeros are complex conjugates of one another.

30
00:05:27,850 --> 00:05:36,230
2i is a known zero of f(x).

31
00:05:36,230 --> 00:06:01,810
The leading coefficient is −3. Find the standard form of f(x).

32
00:06:05,190 --> 00:06:15,590
We know one complex zero. It's imaginary. And we know the leading coefficient.

33
00:06:16,200 --> 00:06:18,933
We know one complex zero. It's imaginary.

34
00:06:20,360 --> 00:06:21,360
Okay.

35
00:06:24,020 --> 00:06:26,380
So what does that tell us?

36
00:06:26,860 --> 00:06:44,740
Its complex conjugate is also a zero. The complex conjugate of 2i is also a zero.

37
00:06:45,740 --> 00:06:49,760
And what is the complex conjugate of two i?

38
00:06:51,900 --> 00:07:03,800
0 + 2i. Its conjugate is 0 − 2i, so −2i is also a zero.

39
00:07:09,660 --> 00:07:13,898
And now we're going to write it in what form? Factored form. When we know the zeros,

40
00:07:13,898 --> 00:07:15,698
we can write factored form.

41
00:07:17,200 --> 00:07:19,400
So it's going to be

42
00:07:21,480 --> 00:07:40,057
f(x) is the leading coefficient, −3, times (x − 2i)(x − (−2i)), which is (x − 2i)(x

43
00:07:40,057 --> 00:07:41,400
+ 2i).

44
00:07:46,720 --> 00:07:52,179
Now we just check. Okay. So that is the function, but it's written in factor form.

45
00:07:52,179 --> 00:07:55,645
And how are we going to get that into standard form?

46
00:07:55,900 --> 00:07:57,160
Distribute.

47
00:07:57,950 --> 00:08:01,630
Minus 4i².

48
00:08:06,170 --> 00:08:10,850
Those two terms add to zero, and i² is −1.

49
00:08:16,070 --> 00:08:18,430
So the product is x² + 4.

50
00:08:20,070 --> 00:08:22,603
And that's not quite in standard form.

51
00:08:24,690 --> 00:08:28,450
f(x) = −3x² − 12.

52
00:08:29,730 --> 00:08:31,270
And there it is.

53
00:08:31,850 --> 00:08:41,450
I could ask other questions: find the vertex, graph it, or what else?

54
00:08:47,530 --> 00:08:50,650
The axis of symmetry. That one is easy.

55
00:08:51,320 --> 00:08:57,442
I could have done no work. What's the axis of symmetry at the very beginning? Zero.

56
00:08:57,442 --> 00:09:00,540
Zero, because it's this number over two a.

57
00:09:01,140 --> 00:09:07,325
Let's try another one slightly more complicated. What was that first one I did? What

58
00:09:07,325 --> 00:09:08,325
was it?

59
00:09:09,220 --> 00:09:20,660
3 + 2i is a known zero.

60
00:09:22,140 --> 00:09:23,540
And

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00:09:27,960 --> 00:09:29,360
the

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00:09:33,060 --> 00:09:34,460
vertex.

63
00:09:39,020 --> 00:09:41,620
Let’s use the axis of symmetry.

64
00:09:42,170 --> 00:09:50,010
What do we know is also a zero? 3 − 2i is also a zero.

65
00:09:53,090 --> 00:09:58,023
So we can write factored form. What is the question asking? Good question.

66
00:09:59,870 --> 00:10:01,910
Write the standard form

67
00:10:09,050 --> 00:10:10,090
of f(x).

68
00:10:15,470 --> 00:10:22,330
We know the zeros, so we can almost write the factored form of f(x).

69
00:10:23,590 --> 00:10:30,090
We don’t have a leading coefficient, so leave an a there for now.

70
00:10:32,750 --> 00:10:37,550
It is x − (3 + 2i)

71
00:10:39,110 --> 00:10:42,670
times x − (3 − 2i).

72
00:10:46,350 --> 00:10:48,270
Let's multiply this out.

73
00:10:49,790 --> 00:11:02,330
We will figure out a. For now: a times (x − 3 − 2i) times (x − 3 + 2i).

74
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Then multiply it all out. There is a lot of distributing. How many terms? Nine.

75
00:11:17,530 --> 00:11:23,870
You could do it. X times X is X squared. X squared.

76
00:11:25,430 --> 00:11:32,310
−3x + 2ix.

77
00:11:37,670 --> 00:11:39,890
Minus three X.

78
00:11:42,930 --> 00:11:49,410
−3x + 9 −6i.

79
00:11:57,690 --> 00:12:04,390
−2ix + 6i

80
00:12:06,450 --> 00:12:08,810
−4i².

81
00:12:14,150 --> 00:12:19,850
Everything with i cancels, or i² turns into −1.

82
00:12:21,070 --> 00:12:31,445
2ix and −2ix cancel. If they didn’t, we multiplied something incorrectly. We have a

83
00:12:31,445 --> 00:12:32,570
times x²…

84
00:12:34,050 --> 00:12:37,590
−3x − 3x is −6x,

85
00:12:39,530 --> 00:12:42,730
and 9 + 4

86
00:12:44,390 --> 00:12:45,300
is 13.

87
00:12:45,550 --> 00:12:48,483
I don't think I gave you enough information.

88
00:12:52,010 --> 00:12:55,070
I don't think so. I messed up.

89
00:12:56,390 --> 00:12:58,450
There are actually many possibilities.

90
00:12:58,700 --> 00:13:05,240
This is ax² − 6ax + 13a. We agree.

91
00:13:11,560 --> 00:13:12,880
So we know.

92
00:13:14,540 --> 00:13:17,400
Negative 6 A.

93
00:13:19,760 --> 00:13:20,760
Over.

94
00:13:22,820 --> 00:13:27,460
Two a. This is our negative b— Sorry negative.

95
00:13:30,800 --> 00:13:32,340
This is equal to 3.

96
00:13:32,990 --> 00:13:38,230
No I didn't. I take that back. I screwed up.

97
00:13:39,610 --> 00:13:43,343
We don’t have enough information to answer the question.

98
00:13:45,430 --> 00:13:46,430
Okay. Yeah.

99
00:13:46,450 --> 00:13:50,750
Let’s instead say this is the leading coefficient.

100
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Then we substitute 3 here.

101
00:14:06,130 --> 00:14:13,510
The function would be 3x² − 18x + 39. Okay.

102
00:14:14,090 --> 00:14:21,450
If you are given just one real zero, you cannot determine the other zero.

103
00:14:22,290 --> 00:14:27,230
But given a nonreal complex zero, its conjugate is also a zero,

104
00:14:30,010 --> 00:14:33,510
so we know the other zero.

105
00:14:33,850 --> 00:14:58,630
A quadratic function has vertex (−3, 2) and passes through

106
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(−1, 14).

107
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Find the standard form.

108
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How we can do this.

109
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Okay. They give us the vertex.

110
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We can almost write vertex form; the only missing quantity is the leading

111
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coefficient. So we got to leave this as an A.

112
00:15:29,680 --> 00:15:37,200
x − (−3) is x + 3, squared, plus 2.

113
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And last thing I do is find the leading coefficient. Multiply that out.

114
00:15:47,080 --> 00:15:53,060
f(−1) = 14. That is what this ordered pair means. We plug that in here.

115
00:15:54,220 --> 00:15:56,120
14 equals a

116
00:15:58,100 --> 00:16:04,640
times (−1 + 3)², plus 2. And now we got an equation with just one variable.

117
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And we do a little bit of work.

118
00:16:11,960 --> 00:16:12,960
14 − 2

119
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equals a times 2 squared.

120
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That is 4a.

121
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Divide by 4.

122
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So this is the.

123
00:16:35,540 --> 00:16:38,060
This is the equation in what form.

124
00:16:39,880 --> 00:16:47,040
Vertex form. How do we get standard form? Just start multiplying out.

125
00:16:47,500 --> 00:16:54,680
I will go quickly: x² + 3x + 3x + 9, then plus 2.

126
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Plus 27 plus 2 is 29. Yes.

127
00:17:08,700 --> 00:17:12,920
OK. So if you're given some information.

128
00:17:14,980 --> 00:17:17,700
Suppose I gave you two real zeros

129
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and one ordered pair.

130
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Substituting that point lets us find the leading coefficient.

131
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OK.

132
00:17:32,010 --> 00:17:37,790
Going back last year. Where does.

133
00:17:39,970 --> 00:17:53,270
Where do y = 2x + 3 and y = −3x − 7 intersect?

134
00:17:56,790 --> 00:18:09,090
You might have done problems like this last year. Yes. The others like this.

135
00:18:09,950 --> 00:18:11,283
And we need to find.

136
00:18:12,550 --> 00:18:18,068
Where they intersect. We won’t finish this example, but we will go a little further.

137
00:18:18,068 --> 00:18:20,201
How will we go about doing this.

138
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OK.

139
00:18:25,790 --> 00:18:29,117
Yeah we're not going to try to graph them because it might not need a very nice

140
00:18:29,117 --> 00:18:30,117
point.

141
00:18:30,130 --> 00:18:40,095
We could if they met at a nice point and figured out. We have y = 2x + 3 and y = −3x

142
00:18:40,095 --> 00:18:41,095
− 7.

143
00:18:44,170 --> 00:18:49,910
We want an ordered pair that makes both equations true.

144
00:18:51,790 --> 00:18:55,390
And we have another name for that. What do we call it.

145
00:18:56,670 --> 00:18:59,110
A system of equations.

146
00:19:00,690 --> 00:19:08,730
OK. And last year there were like three ways to solve systems equations.

147
00:19:12,310 --> 00:19:14,530
Elimination, substitution, and…

148
00:19:15,650 --> 00:19:20,050
I think it was the only way.

149
00:19:21,850 --> 00:19:28,710
Yeah. Yeah. Which we don't do. OK. So for this what we do.

150
00:19:31,410 --> 00:19:38,587
Substitute. We take this guy. Substitute in right. This is y, so substitute it for

151
00:19:38,587 --> 00:19:39,587
this y. OK.

152
00:19:40,050 --> 00:19:46,188
We get 2x + 3 = −3x + 7. [Correction on screen: the original constant is −7.] And we

153
00:19:46,188 --> 00:19:49,330
solve this and get some number for X right.

154
00:19:50,210 --> 00:19:56,760
So is that our solution. No. No, that is only the x-coordinate. Substitute x into

155
00:19:56,760 --> 00:19:59,590
either original equation to find y.

156
00:20:00,550 --> 00:20:03,170
So we're going to do something similar but now with quadratics.

157
00:20:03,460 --> 00:20:06,380
We want to find where.

158
00:20:09,220 --> 00:20:10,220
They intersect.

159
00:20:19,600 --> 00:20:26,120
First off. How many possible solutions. How many possible.

160
00:20:29,680 --> 00:20:31,760
Times can intersect.

161
00:20:36,780 --> 00:20:39,040
Does this one open up or down.

162
00:20:43,000 --> 00:20:47,460
So it opens up. This one open up or open down down.

163
00:20:49,420 --> 00:20:54,018
With one opening up and one opening down, how many intersection points could there

164
00:20:54,018 --> 00:20:55,870
be? Two. Can we have just one.

165
00:20:56,160 --> 00:20:59,300
They can meet at a single point. Well.

166
00:21:00,780 --> 00:21:02,647
They could be at the vertex.

167
00:21:03,380 --> 00:21:08,360
OK. And could there be no solutions or no intersections. Yes. Yeah. If this guy was

168
00:21:08,360 --> 00:21:09,360
down here.

169
00:21:09,660 --> 00:21:11,727
So how are we going to do this.

170
00:21:12,500 --> 00:21:23,740
So. OK. Same thing as. Right. y = x² + 2x − 2 and y = −x²

171
00:21:23,740 --> 00:21:26,240
+ 4x + 10.

172
00:21:27,420 --> 00:21:30,220
And this is a system we're going to solve.

173
00:21:30,520 --> 00:21:31,520
OK.

174
00:21:33,020 --> 00:21:37,006
Systems are we are looking for solutions that are common to both of them. So the

175
00:21:37,006 --> 00:21:39,540
solution is going to be ordered pairs.

176
00:21:39,660 --> 00:21:41,260
And we're solve this by.

177
00:21:41,920 --> 00:21:47,478
Substitution. We're going to substitute this. Which is equal to Y into this Y or you

178
00:21:47,478 --> 00:21:49,011
could do it vice versa.

179
00:21:52,500 --> 00:21:59,840
x² + 2x − 2 = −x²

180
00:22:01,400 --> 00:22:03,060
+ 4x + 10.

181
00:22:07,580 --> 00:22:13,500
Oh now we have an equation in only one variable. And now we're going to solve this.

182
00:22:15,380 --> 00:22:19,513
It's just a quadratic equation. We get everything on one side.

183
00:22:19,860 --> 00:22:26,280
Add x², subtract 4x, and subtract 10.

184
00:22:26,900 --> 00:22:36,840
2x² − 2x − 12 = 0.

185
00:22:38,900 --> 00:22:43,760
And then you could solve by factoring completing the square quadratic formula. These

186
00:22:43,760 --> 00:22:46,894
have a common factor. Which should we do first?

187
00:22:47,460 --> 00:22:51,327
The two. Yeah get rid of that two. It will make it easier.

188
00:22:56,100 --> 00:22:57,967
And does this factor easily?

189
00:23:03,840 --> 00:23:05,640
It does. What's the factor?

190
00:23:06,880 --> 00:23:17,960
x − 3 times x + 2. The solutions are 3 and −2.

191
00:23:18,570 --> 00:23:22,910
Is that my answer? 3, −2? Those are X values.

192
00:23:27,590 --> 00:23:34,726
So to find the ordered pairs we plug them into. Either one of them. Let's plug it in

193
00:23:34,726 --> 00:23:37,190
here. What was the first one?

194
00:23:39,490 --> 00:23:46,090
f(3). 3² + 2 times 3 − 2.

195
00:23:51,290 --> 00:23:53,810
9 + 6 − 2

196
00:23:55,810 --> 00:23:56,810
is 13.

197
00:23:59,370 --> 00:24:06,710
And g of… What was the other one? Negative one? Negative two. Negative two.

198
00:24:24,170 --> 00:24:29,450
Is that negative two? No. So they intersect.

199
00:24:34,110 --> 00:24:37,890
(3, 13)

200
00:24:40,930 --> 00:24:42,350
and (−2,

201
00:24:44,190 --> 00:24:45,470
−2).

202
00:24:46,590 --> 00:24:47,470
Okay.

203
00:24:47,720 --> 00:24:52,021
We could have a quadratic and a linear. Do the same thing. Substitute the linear

204
00:24:52,021 --> 00:24:55,021
expression into the quadratic, or vice versa.

205
00:24:55,700 --> 00:25:00,567
And we solve that quadratic. If you end up with the irrational solutions.

206
00:25:02,220 --> 00:25:06,860
Substitution into the functions may look less tidy.

207
00:25:07,340 --> 00:25:15,400
Find the equation of the line

208
00:25:20,980 --> 00:25:22,380
That.

209
00:25:26,380 --> 00:25:27,780
Passes.

210
00:25:32,400 --> 00:25:33,400
through

211
00:25:41,140 --> 00:25:42,140
Both.

212
00:25:43,520 --> 00:25:44,520
vertices.

213
00:25:45,130 --> 00:25:52,650
The quadratic has a vertex, and the absolute-value function has a vertex.

214
00:25:56,270 --> 00:25:59,570
What is the Vertex of this one? This one? easy.

215
00:26:04,370 --> 00:26:11,571
(3, 1). This is in vertex form. Take the opposite of this number, paired with that

216
00:26:11,571 --> 00:26:15,610
number. How do we find the vertex of this one?

217
00:26:25,070 --> 00:26:31,050
−2. −2. And now we plug it in to find the y value.

218
00:26:42,250 --> 00:26:47,030
4 − 8 + 1. Is that −3? Yes.

219
00:26:48,330 --> 00:26:50,870
Okay, so the vertex here is

220
00:26:53,450 --> 00:26:59,730
(−2, −3). And then what did I ask for?

221
00:27:02,630 --> 00:27:05,763
The equation of the line through both vertices.

222
00:27:09,150 --> 00:27:12,750
I don't want to graph it. I want to find the equation.

223
00:27:15,050 --> 00:27:18,430
Okay, we need to find the slope first.

224
00:27:19,890 --> 00:27:25,470
Which is going to be right. (y₂ − y₁)/(x₂ − x₁).

225
00:27:27,190 --> 00:27:37,470
1 minus negative 3, over 3 minus negative 2. That is 4/5.

226
00:27:38,890 --> 00:27:44,150
Now we need the y value. We can then find the y intercept or

227
00:27:45,470 --> 00:27:50,530
The better way is point-slope form. Oh, yeah.

228
00:27:54,710 --> 00:27:58,110
Which point should I use? This point or this point?

229
00:28:00,010 --> 00:28:03,190
I think that was the point. It's nice.

230
00:28:05,330 --> 00:28:08,630
It doesn't matter.

231
00:28:10,110 --> 00:28:15,910
y − 1 = (4/5) times ( x

232
00:28:19,650 --> 00:28:20,670
minus 3).

233
00:28:24,150 --> 00:28:25,150
And that's it.

234
00:28:27,610 --> 00:28:32,162
Do I need to put it in slope-intercept form? No, it is already the equation of the

235
00:28:32,162 --> 00:28:34,429
line. No matter what form it's in.

236
00:28:36,330 --> 00:28:40,350
So we can answer all kinds of crazy questions now.